Inertia weight control strategies for particle swarm optimization

Abstract

Particle swarm optimization (PSO) is a population-based, stochastic optimization technique inspired by the social dynamics of birds. The PSO algorithm is rather sensitive to the control parameters, and thus, there has been a significant amount of research effort devoted to the dynamic adaptation of these parameters. The focus of the adaptive approaches has largely revolved around adapting the inertia weight as it exhibits the clearest relationship with the exploration/exploitation balance of the PSO algorithm. However, despite the significant amount of research efforts, many inertia weight control strategies have not been thoroughly examined analytically nor empirically. Thus, there are a plethora of choices when selecting an inertia weight control strategy, but no study has been comprehensive enough to definitively guide the selection. This paper addresses these issues by first providing an overview of 18 inertia weight control strategies. Secondly, conditions required for the strategies to exhibit convergent behaviour are derived. Finally, the inertia weight control strategies are empirically examined on a suite of 60 benchmark problems. Results of the empirical investigation show that none of the examined strategies, with the exception of a randomly selected inertia weight, even perform on par with a constant inertia weight.

Appendices

Appendix 1: Benchmark problems

This appendix provides a further, in-depth definition of the benchmark problems employed in this study.

A function, f, was shifted using

$$\begin{aligned} f^\text {{Sh}}(\mathbf x ) = f(\mathbf x -\gamma ) + \beta \end{aligned}$$

where $\beta$ and $\gamma$ are constants. Rotation was implemented using either a randomly generated orthonormal rotation matrix, denoted by ‘ortho’, or a linear transformation matrix, denoted by ‘linear’. In either scenario, the rotation was performed using Salomon’s method (Salomon 1996) with a new rotation matrix computed for each of the independent runs using the condition number provided. The rotated functions, denoted by $f^\text {{R}}$, were then computed by multiplying the decision vector $\mathbf x$ by the transpose of the rotation matrix. Noisy functions, denoted by $f^\text {N}$, were generated by multiplying each decision variable by a noise value sampled from a Gaussian distribution with the specified mean and deviation. Table 2 provides the configuration parameters for the shifted, rotated, rotated and shifted, and noisy versions of the functions in the columns ‘Sh’, ‘R’, ‘ShR’, and ‘N’, respectively.

It should be noted that the composition functions $f_{27} - f_{37}$ are equal to functions $f_{15} - f_{25}$ from the CEC 2005 benchmark set (Suganthan et al. 2005). For each of these problems, the composed functions are each rotated using linear transformation matrices with independent condition numbers. Thus, the reader is directed to Suganthan et al. (2005) for specific details regarding the configurations of these functions.

The equations for each benchmark problem are as follows.

$f_1,$ :

the absolute value function, defined as

$$\begin{aligned} f_1(\mathbf x ) = \sum _{j=1}^{n_x} |x_j| \end{aligned}$$

(38)

with each $x_j \in [-100,100]$.

$f_2,$ :

the Ackley function, defined as

$$\begin{aligned} f_2(\mathbf x ) = -20e^{-0.2\sqrt{\frac{1}{n}\sum _{j=1}^{n_x} x_j^2}} - e^{\frac{1}{n}\sum _{j=1}^{n_x}\cos (2 \pi x_j)} + 20 + e \end{aligned}$$

(39)

with each $x_j \in [-32.768, 32.768]$. Shifted, rotated, and rotated and shifted versions of $f_{2}$ were also used.

$f_3,$ :

the alpine function, defined as

$$\begin{aligned} f_3(\mathbf x ) = \left( \prod _{j=1}^{n_x} \sin (x_j) \right) \sqrt{\prod _{j=1}^{n_x} x_j} \end{aligned}$$

(40)

with each $x_j \in [-10, 10]$.

$f_4,$ :

the egg holder function, defined as

$$\begin{aligned} f_4(\mathbf x ) = \sum _{j=1}^{n_x - 1} \left( -(x_{j+1} + 47)\sin \left( \sqrt{|x_{j+1} + x_j/2 + 47|}\right) + \sin \left( \sqrt{|x_j - (x_{j+1} + 47)| }\right) (-x_j) \right) \end{aligned}$$

(41)

with each $x_j \in [-512,512]$.

$f_5,$ :

the elliptic function, defined as

$$\begin{aligned} f_5(\mathbf x ) = \sum _{j=1}^{n_x} (10^6)^\frac{j-1}{n_x-1} \end{aligned}$$

(42)

with each $x_j \in [-100, 100]$. Shifted, rotated, and rotated and shifted versions of $f_{5}$ were also used.

$f_6,$ :

the Griewank function, defined as

$$\begin{aligned} f_6(\mathbf x ) = 1 + \frac{1}{4000} \sum _{j=1}^{n_x} x_j^2 - \prod _{j=1}^{n_x} \cos \left( \frac{x_j}{\sqrt{j}} \right) \end{aligned}$$

(43)

with each $x_j \in [-600,600]$. Shifted, rotated, and rotated and shifted versions of $f_{6}$ were also used. For the rotated and shifted version of $f_6$, the range was modified to $x_j \in [0, 600]$ such that the global minimum was outside the bounds.

$f_7,$ :

the hyperellipsoid function, defined as

$$\begin{aligned} f_7(\mathbf x ) = \sum _{j=1}^{n_x} j x_j^2 \end{aligned}$$

(44)

with each $x_j \in [-5.12,5.12]$.

$f_8,$ :

the Michalewicz function, defined as

$$\begin{aligned} f_8(\mathbf x ) = -\sum _{j=1}^{n_x} \sin (x_j)\left( \sin \left( \frac{j x_j^2}{\pi } \right) \right) ^{2m} \end{aligned}$$

(45)

with each $x_j \in [0,\pi ]$ and $m=10$.

$f_9,$ :

the norwegian function, defined as

$$\begin{aligned} f_9(\mathbf x ) = \prod _{j=1}^{n_x} \left( \cos (\pi x_j^3) \left( \frac{99+x_j}{100} \right) \right) \end{aligned}$$

(46)

with each $x_j \in [-1.1,1.1]$.

$f_{10},$ :

the quadric function, defined as

$$\begin{aligned} f_{10}(\mathbf x ) = \sum _{i=1}^{n_x} \left( \sum _{j=1}^{i} x_j \right) ^2 \end{aligned}$$

(47)

with each $x_j \in [-100,100]$.

$f_{11},$ :

the quartic function, defined as

$$\begin{aligned} f_{11}(\mathbf x ) = \sum _{j=1}^{n_x} j x_j^4 \end{aligned}$$

(48)

with each $x_j \in [-1.28,1.28]$. A noisy version of the quartic function, referred to as De Jong’s f4 function, was generated according to

$$\begin{aligned} f_{11}^N(\mathbf x ) = \sum _{j=1}^{n_x} (j x_j^4 + N(0, 1)) \end{aligned}$$

(49)

using the same domain as the quartic function.

$f_{12},$ :

the Rastrigin function, defined as

$$\begin{aligned} f_{12}(\mathbf x ) = 10n_x + \sum _{j=1}^{n_x} (x_j^2 - 10\cos (2 \pi x_j)) \end{aligned}$$

(50)

with each $x_j \in [-5.12,5.12]$. Shifted, rotated, and rotated and shifted versions of $f_{12}$ were also used.

$f_{13},$ :

the Rosenbrock function, defined as

$$\begin{aligned} f_{13}(\mathbf x ) = \sum _{j=1}^{n_x-1} \left( 100(x_{j+1} - x_j^2) + (x_j - 1)^2\right) \end{aligned}$$

(51)

with each $x_j \in [-30,30]$. Shifted and rotated versions of $f_{13}$ were also used. Both the rotated and shifted versions of the $f_{13}$ used the domain $[-100,100]$.

$f_{14},$ :

the Saloman function, defined as

$$\begin{aligned} f_{14}(\mathbf x ) = -\cos \left( 2 \pi \sum _{j=1}^{n_x} x_j^2 \right) + 0.1 \sqrt{\sum _{j=1}^{n_x} x_j^2} + 1 \end{aligned}$$

(52)

with each $x_j \in [-100,100]$.

$f_{15},$ :

the Schaffer 6 function, defined as

$$\begin{aligned} f_{15}(\mathbf x ) = \sum _{j=1}^{n_x} \left( 0.5 + \frac{\sin ^2(x_j^2 + x_{j+1}^2) - 0.5}{(1+0.001(x_j^2 + x_{j+1}^2))^2} \right) \end{aligned}$$

(53)

with each $x_j \in [-100,100]$. A rotated and shifted version of $f_{15}$ was also used.

$f_{16},$ :

the Schwefel 1.2 function, defined as

$$\begin{aligned} f_{16}(\mathbf x ) = \sum _{i=1}^{n_x}\left( \sum _{j=1}^{i} x_j\right) ^2 \end{aligned}$$

(54)

with each $x_j \in [-100,100]$. Shifted and rotated versions of $f_{16}$ were also used. Additionally, a shifted, noisy version of the Schwefel 1.2 function, defined as

$$\begin{aligned} f_{16}^{ShN}(\mathbf x ) = \sum _{i=1}^{n_x}\left( \sum _{j=1}^{i} x_j\right) ^2 (1 + 0.4|N(0,1)|) \end{aligned}$$

(55)

was also used, with the same domain as the base function.

$f_{17},$ :

the Schwefel 2.6 function, defined as

$$\begin{aligned} f_{17}(\mathbf x ) = \max _j\{|\mathbf A _j\mathbf x - \mathbf B _j|\} \end{aligned}$$

(56)

with each $x_j \in [-100,100]$, each $a_{ij} \in \mathbf A$ is uniformly sampled from $U(-500,500)$ such that $\det (\mathbf A ) \ne 0$, and each $\mathbf B _j = \mathbf A _j\mathbf r$ where each $r_i \in \mathbf r$ is uniformly sampled from $U(-100,100)$. A shifted version of $f_{17}$ was also used.

$f_{18},$ :

the Schwefel 2.13 function, defined as

$$\begin{aligned} f_{18}(\mathbf x ) = \sum _{j=1}^{n_x} (\mathbf A _j - \mathbf B _j(\mathbf x ))^2 \end{aligned}$$

(57)

with each $x_j \in [-\pi ,\pi ]$, and

$$\begin{aligned} \mathbf A _j = \sum _{i=1}^{n_x}(a_{ij}\sin (\alpha _i) + b_{ij}\cos (\alpha _i)) \end{aligned}$$

and

$$\begin{aligned} \mathbf B _j(\mathbf x ) = \sum _{i=1}^{n_x}(a_{ij}\sin (x_i) + b_{ij}\cos (x_i)) \end{aligned}$$

where $a_{ij} \in \mathbf A , b_{ij} \in \mathbf B$, $a_{ij}, b_{ij} \sim U(-100,100)$, and $\alpha _i \sim U(-\pi , \pi )$. A shifted version of $f_{18}$ was also used.

$f_{19},$ :

the Schwefel 2.21 function, defined as

$$\begin{aligned} f_{19}(\mathbf x ) = \max _j\{|x_j|, 1 \le j \le n_x\} \end{aligned}$$

(58)

with each $x_j \in [-100,100]$.

$f_{20},$ :

the Schwefel 2.22 function, defined as

$$\begin{aligned} f_{20}(\mathbf x ) = \sum _{j=1}^{n_x}|x_j| + \prod _{j=1}^{n_x}|x_j| \end{aligned}$$

(59)

with each $x_j \in [-10,10]$.

$f_{21},$ :

the Shubert function, defined as

$$\begin{aligned} f_{21}(\mathbf x ) = \prod _{j=1}^{n_x}\left( \sum _{i=1}^{5} (i\cos ((i+1)x_j + i)) \right) \end{aligned}$$

(60)

with each $x_j \in [-10,10]$.

$f_{22},$ :

the spherical function, defined as

$$\begin{aligned} f_{22}(\mathbf x ) = \sum _{j=1}^{n_x} x_j^2 \end{aligned}$$

(61)

with each $x_j \in [-5.12,5.12]$. A shifted version of $f_{22}$ was also used.

$f_{23},$ :

the step function, defined as

$$\begin{aligned} f_{23}(\mathbf x ) = \sum _{j=1}^{n_x} (\lfloor x_j + 0.5 \rfloor )^2 \end{aligned}$$

(62)

with each $x_j \in [-100,100]$.

$f_{24},$ :

the Vincent function, defined as

$$\begin{aligned} f_{24}(\mathbf x ) = - \left( 1 + \sum _{j=1}^{n_x} \sin (10 \sqrt{x_j})\right) \end{aligned}$$

(63)

with each $x_j \in [0.25,10]$.

$f_{25},$ :

the Weierstrass function, defined as

$$\begin{aligned} \begin{aligned} f_{25}(\mathbf x ) =&\sum _{j=1}^{n_x}\left( \sum _{i=1}^{20} (a^i\cos (2 \pi b^i(x_j + 0.5))) \right) \\&-n_x\sum _{i=1}^{20}(a^i\cos (\pi b^i)) \end{aligned} \end{aligned}$$

(64)

with each $x_j \in [-0.5,0.5], a=0.5$, and $b=3$. A rotated and shifted version of $f_{25}$ was also used.

$f_{26},$ :

a shifted expansion of the Griewank and Rosenbrock functions [Eqs. (43) and (51), respectively] with each $x_j \in [-3,1]$. Note that $f_{26}$ is equivalent to $f_{13}$ from the 2005 CEC benchmark suite.

$f_{27} - f_{37},$ :

composition functions equivalent to $f_{15} - f_{25}$ from the 2005 CEC benchmark suite. All functions have each $x_j \in [-5,5]$, with the exception of $f_{37}$ which has each $x_j \in [2,5]$.

Appendix 2: Overall ranks using the local-best topology

Table 11 summarizes the results obtained across all benchmark problems using the Mann–Whitney U statistical analysis procedure described in Sect. 5.3. Table 11 clearly indicates that the influence of topology is significant with regard to the performance of the inertia weight strategies. Specifically, the constant and random strategies are no longer the top performing strategies when the local-best topology is considered. Rather, the constant strategy attained a rank of 5 and the random strategy attained a rank of 9 when considering the accuracy performance measure. Despite its relatively poor rank of 13 when using the global-best topology, the PSO-NL strategy showed the best overall accuracy when using the local-best topology. Thus, the poor performance of the adaptive inertia weight strategies observed when the global-best topology was used, does not necessarily hold when the local-best topology is used. Therefore, it can be concluded that the topology should be considered when tuning for optimal performance. This comes as no surprise given that it is well known that the topology should be included as a tuned parameter if optimal performance is desired Engelbrecht (2013a).

Table 11 Summary of performance across all 60 benchmark problems using the local-best topology